OFFSET
0,4
COMMENTS
The number of all noncrossing caterpillars with n edges is given by A361356.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-3,-26,36,-2,-18,6,5,-4,1).
FORMULA
G.f.: (1 - 5*x - 2*x^2 + 27*x^3 - 20*x^4 - 13*x^5 + 23*x^6 - 5*x^7 - 6*x^8 + 3*x^9)/((1 - x)*(1 - 5*x + 3*x^2 - x^3)*(1 - 5*x^2 + 3*x^4 - x^6)).
a(n) = 6*a(n-1) - 3*a(n-2) - 26*a(n-3) + 36*a(n-4) - 2*a(n-5) - 18*a(n-6) + 6*a(n-7) + 5*a(n-8) - 4*a(n-9) + a(n-10) for n >= 10.
PROG
(PARI)
G(x)={ my(f = x*(2 - x)/(1 - 5*x + 3*x^2 - x^3), g = 1 + x + x^2*(3 - 2*x + (4 - 3*x + x^2)*f + (1 + 2*x)*f^2)/(1 - x)^2); (intformal(g) - 3)/x + x*subst((1 + 2*x*f)/(1-x)^2, x, x^2)/2 }
{ Vec(G(x) + O(x^30)) }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, Mar 09 2023
STATUS
approved